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Full text of "Treatise On Analysis Vol-Ii"

436 XV NORMED ALGEBRAS AND SPECTRAL THEORY for n 2> 0, which are all strictly positive, show that P0(£) == 1 and that (3) - '^ l C2 •*• Cn + 1 Cn *•• C2n-i 1 for all n > 1. Also we have (4) (e) Conversely, for every (bounded or unbounded) positive measure ^ on R, show that the operator MM (which is in general unbounded) is self-adjoint and simple. 2. Conversely, let / be a Jacobi matrix (formula (1) of Problem 1) in which the an are real and the bn are real and >0, but are not restricted in any other way. Let (en)n^o be the canonical Hilbert basis of the Hilbert space /£ (6.5). Define a hermitian operator H, with domain the subspace G of /£ generated by the (finite) linear com- binations of the en, by the conditions H- en = bn-i en-i + anen + bnen^ (n ^0) (with e~i = 0, £_! = 0). By abuse of notation, let H also denote the closure of the hermitian operator H. 00 (a) Let £ be a complex number. If y = £ yn en is an eigenvector of H* corresponding to the eigenvalue £, show that yn = Pn(QyG, where Pn is a polynomial of degree /z, determined recursively by the formulas PO(£) = 1 and for all n 2> 0. Deduce that the defect of His (1, 1) if, for some £ e C such that ./£ ^ 0, we have 00 ~~ " ; < + oo: in which case this inequality is true for all £ e C such that ./£ =£ 0. In the contrary case, the defect of H is (0,0), in other words H is self-adjoint (having regard to the abuse of language mentioned above). In every case there exist self-adjoint extensions ofH. (b) Let F be the subspace of ^C(R) consisting of polynomials with complex coeffi- cients. Show that there exists a unique linear form ap on F such that ap(Pn Pm) = $mn (Kronecker index). (If a polynomial is of the form RS, where R = ]T ut Pt and S = ]T vt PI , then we must have aF(RS) =2 «* ty, which shows that aF is unique. For the existence, it is enough to show that, if we write the product U(jc) = (x — z0)R(jc)S(x) in the two forms ((* — z0)R(x))S(x) and R(#)((# — z0)S(x)), where z0 is any complex number, then the calculation of aF(U) from each of these two products must give the same value. Reduce to the case where each of R, S is one of the polynomials Pn, and use the relation (1).) (15.12.8),